Underwater Robots - Gianluca Antonelli.pdf

8.4 External Force Control 205
The second-order differential relationship between the linear end-effector
velocity inthe earth-fixed frame and the system velocity isgiven by
¨x = J pos( q , R I B ) ˙ ζ + ˙ J pos( q , R I B ) ζ
that, neglecting dependencies, can be inverted inthe simplest way as
�
¨x − ˙ �
J posζ
˙ζ = J † pos
that, plugged into (8.2), and defining
τ = J T posf (8.3)
where f ∈ IR 3 is to be defined, leads to
M ¨x + f n = f − f e
(8.4)
where M = J T posMJ † pos, and f n = J T posτ n − M ˙ J posJ † pos ˙x .The vector
f ∈ IR 3 can be selected as
where
f = ˆ Mxc + f n + f e . (8.5)
− 1
x c = ¨x d + M d
�
D d ˙˜x
�
+ K d ˜x − f e
(8.6)
Details can be found in [94] as well as [254]; Figure 8.2 represents ablock
scheme of the impedance approach. In[95, 96] aunified version with the
hybrid approach isalso proposed.
x p,d, x s,d
η , q
eq. (8.6)
¨x c +
Fig. 8.2. Impedance Control Scheme
8.4 External Force Control
+
+
+
f
J T pos
τ −
+
f n
J T pos
UVMS
+env.
In this Section aforce control scheme is presented to handle the strong limitations
that are experienced in case of underwater systems. Based on the
f e